Existence of solutions for some noncoercive elliptic problems involving derivatives of nonlinear terms (Q2885123)

Let \(\Omega \subset \mathbb R^n\), \(n \geq 3\), be bounded and open; \(a\) and \(b\) measurable bounded non-negative functions on \(\Omega\) with \(a(x) \geq \alpha > 0\) bounded away from zero. For \(\Phi: \mathbb R \to \mathbb R^n\) continuous, and \(f \in L^2(\Omega)\), the authors pose the degenerate elliptic problem NEWLINE\[NEWLINE - \mathrm{div} \left( \frac{a(x) \nabla u}{(1+b(x)|u|)^2}\right) + u = f - \mathrm{div}(\Phi(u)) \tag{1} NEWLINE\]NEWLINE in the Sobolev space \(W_0^{1,1}(\Omega)\). Based on a priori estimates for a regularising sequence of problems (with \(f_n\) in the right-hand side of (1) essentially bounded on \(\Omega\)), the authors prove that if \(|\Phi(t)| \leq C t^2\) for some constant \(C\) and all \(t \in \mathbb R\), then there exists a solution \(u \in W_0^{1,1}(\Omega) \cap L^2(\Omega)\) in the sense that NEWLINE\[NEWLINE \int_\Omega \frac{a(x) \nabla u \cdot \nabla \varphi}{(1+b(x)|u|)^2}\,dx + \int_{\Omega} u \varphi \, dx = \int_{\Omega} f \varphi \, dx + \int_{\Omega} \Phi(u) \cdot \nabla \varphi \, dx, NEWLINE\]NEWLINE for all test functions \(\varphi \in W_0^{1, \infty}\). In the case when \(\Phi\) is only continuous, so does not satisfy a growth condition, the authors establish the existence of an entropy (or renormalised) solution in the same function class.